The number of signals that can be generated by using 6 differently coloured flags, when any number of them may be hoisted at a time is

Options

A. 1956
B. 1957
C. 1958
D. 1959

Explanation

To solve the problem of how many signals can be generated using 6 differently colored flags when any number of them may be hoisted at a time, we can follow these steps:

### Step 1: Understanding the Problem
We need to determine how many different signals can be created using 6 flags, where we can use any number of flags from 1 to 6.

### Step 2: Counting Arrangements
For each possible number of flags (R), we can choose and arrange them. The number of ways to choose R flags from 6 is given by the combination formula \( \binom{n}{r} \), and the number of ways to arrange R flags is given by \( R! \).

### Step 3: Setting Up the Summation
We need to sum the arrangements for all possible values of R from 1 to 6:
\[ \text{Total Signals} = \sum_{R=1}^{6} \binom{6}{R} \times R! \]

### Step 4: Calculating Each Term
Now we calculate each term in the summation:
- For \( R = 1 \):
\[ \binom{6}{1} \times 1! = 6 \times 1 = 6 \]

- For \( R = 2 \):
\[ \binom{6}{2} \times 2! = 15 \times 2 = 30 \]

- For \( R = 3 \):
\[ \binom{6}{3} \times 3! = 20 \times 6 = 120 \]

- For \( R = 4 \):
\[ \binom{6}{4} \times 4! = 15 \times 24 = 360 \]

- For \( R = 5 \):
\[ \binom{6}{5} \times 5! = 6 \times 120 = 720 \]

- For \( R = 6 \):
\[ \binom{6}{6} \times 6! = 1 \times 720 = 720 \]

### Step 5: Summing All Terms
Now we add all the calculated values:
\[ 6 + 30 + 120 + 360 + 720 + 720 = 1956 \]

### Final Answer
Thus, the total number of signals that can be generated by using 6 differently colored flags is **1956**.

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